Why are you mentionning "radius" in "title", but no reference to radius in the problem statement?
With your example (legs 4 and 8), the chord that supports the arc is the hypotenuse,
so 4SQRT(5) = ~8.944 ; agree?
The radius that makes that possible (arc = ~9.342) = ~9.182, and supporting angle = ~58.308 deg.
I believe this can only be obtained by numeric methods; that what I used anyway: ye olde brute force!
I'll wait for your reply: is anything else (such as radius) given?
Do you agree with my radius and angle calculation above?